Multirate Time Integration Research Articles

Multirate time-integration based on dynamic ODE partitioning through adaptively refined meshes for compressible fluid dynamics

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1,2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

Aeroelastic simulations of the DTU 10 MW turbine using tight coupling integration techniques

The present study aims to investigate a newly developed multirate time integration technique on aeroelastic simulations of the DTU 10 MW reference wind turbine inside a turbulent atmospheric boundary layer. The turbine is modelled using an actuator line model and a multibody structural solver is employed for the dynamic response. The turbine is studied under cases of laminar flow and a pressure driven boundary layer. Results show that the scheme exhibits high accuracy and matches the theoretical order of the Runge–Kutta scheme. During unsteady operations, substantial deformations were observed that are influenced mainly by the turbine’s rotation and the effect of turbulence in fatigue loading is highlighted.

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

Nonlocal thermomechanical coupled modeling method for two-dimensional rolling contact using a peridynamic approach

The development of thermomechanical coupled simulations for rolling contacts is challenging when the adjacent material is temperature-dependent or when there are cracks or other discontinuities on the contact surface. This paper proposes a novel thermomechanical coupled modeling method for rolling contacts using a nonlocal peridynamic (PD) approach. Fully coupled, ordinary state-based PD thermomechanical equations for the plane strain and plane stress cases are derived based on the strain-energy density, including thermal coupling terms, and detailed information on each parameter is provided. Subsequently, an algorithm for rolling thermal contacts in PD is proposed based on the penalty method. This study further provides a detailed modeling process and an algorithm for the proposed method based on a multi-rate time-integration scheme for numerical implementation. A typical two-dimensional (2D) wheel–rail rolling contact model is established to evaluate the performance of the proposed contact modeling method in wheel–rail thermal contact and to analyze the influence of temperature-dependent material and rail-surface cracks on the thermomechanical response. It is found that the contact stress and rail-surface temperature rise in the case of small creepages and temperature-independent materials are in good agreement with the analytical solution. In the case of large creepages, the mechanical and temperature results of using temperature-dependent materials and temperature-independent materials are quite different, and the frictional heat generated can cause martensitic transformation of adjacent materials. Rail-surface cracks cause a sharp increase in the contact stress and temperature rise near the cracks, and the high temperature also promotes crack growth.

An assessment of the utility of multirate time integration for the modeling of laser powder bed fusion

An assessment of the utility of multirate time integration for the modeling of laser powder bed fusion

Multirate DAE-simulation and its application in system simulation software for the development of electric vehicles

This work is devoted to the efficient simulation of large multi-physical networks stemming from automated modelling processes in system simulation software. The simulation of hybrid, battery and fuel cell electric vehicle applications requires the coupling of electrical, mechanical, fluid, and thermal networks. Each network is established by combining the connection structure of a graph with physical equations of elementary components and resulting in a DAE (differential algebraic equation). To speed up the simulation a non-iterative multirate time integration co-simulation method for the system of coupled DAEs is established. The power of the multirate method is shown via two representative examples of battery powered electric vehicle with a cooling system for the battery pack and a three phase inverter with a cooling system. This work is an extended version of (Kolmbauer et al. in Scientific Computing in Electrical Engineering (SCEE 2020), Springer, Cham, pp. 231–240, 2021).

Offline adaptation of co-simulation time steps by leveraging past co-simulation runs in multi-level mechatronic systems

Modeling and simulation of mechatronic systems require a multi-disciplinary approach as several physical domains interact with each other. Due to this heterogeneity, each domain can benefit from a dedicated time integration strategy as well as modeling formalism to control the error in simulation and reduce computational overhead. Current multi-rate adaptive time-stepping approaches disregard the system dynamics knowledge obtained in previous simulations and adapt the time steps at run-time as a black-box model. Although current multi-rate adaptive time-stepping approaches do not exploit the time scale of the physics of each domain, this information can be exploited in co-simulations by having a predefined multi-rate time integration approach in design space exploration and parameter identification applications, which involve repeated simulations with similar dynamics. In this study, an offline adaptive co-simulation technique is introduced. The proposed method adapts the communication and time integration time steps before the simulation using the estimated error data from past co-simulation runs. The offline adaptive co-simulations performed faster against the run-time adaptive co-simulations for comparable accuracy levels in 1D and multi-level mechatronic systems with changing loading and initial conditions.

Multirate linearly-implicit GARK schemes

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

A conservative multirate explicit time integration method for computation of compressible flows

In the context of high fidelity simulation of compressible flows (LES and DNS) at extreme scale (small time steps) on massively parallel supercomputers, explicit time integration methods are widely used since they both have a good computational cost trade-off at small time scales and require a small number of parallel communications, thus having little impact on the parallel strategy compared to their implicit counterpart. However, the synchronous nature of the time integration of the governing equations can be a severe constraint when the stability condition is applied globally since the time scales are related to the flow properties and the mesh resolution, which may show strong variations throughout the computational domain. We propose to further improve the efficiency (CPU wall-time reduction) of explicit Runge–Kutta methods by developing a multirate explicit time integration method, by means of flux interpolation at the boundary between cells evolving with different time-steps, which enforces the conservation properties. In terms of computational efficiency, the presented multirate time integration method is easy to implement in pre-existing Eulerian compressible Navier–Stokes codes, requires less additional memory storage, and provides a considerable speed-up while being robust and preserving the order of accuracy of the legacy explicit Runge–Kutta time integration method. The multirate time integration method is implemented in the massively parallel finite volume and high-order (spectral difference) IC3 code (Bodart et al., 2016) (fork of the solver CharLESX), but it can also be applied to any flux-based spatial method such as discontinuous Galerkin or others. For a targeted y+=0.2 on the developed turbulent channel flow test case at Reτ=392; a 2.48 effective speedup is obtained versus an expected theoretical speedup of 2.53.

Reduced order multirate schemes for coupled differential-algebraic systems

In the context of time-domain simulation of integrated circuits, one often encounters large systems of coupled differential-algebraic equations. Simulation costs of these systems can become prohibitively large as the number of components keeps increasing. In an effort to reduce these simulation costs a twofold approach is presented in this paper. We combine maximum entropy snapshot sampling method and a nonlinear model order reduction technique, with multirate time integration. The obtained model order reduction basis is applied using the Gauß-Newton method with approximated tensors reduction. This reduction framework is then integrated using a coupled-slowest-first multirate integration scheme. The convergence of this combined method is verified numerically. Lastly it is shown that the new method results in a reduction of the computational effort without significant loss of accuracy.

Implicit-Explicit Multirate Infinitesimal GARK Methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

Towards improved speed and accuracy of laser powder bed fusion simulations via representation of multiple time scales

The current work reports the result of investigations into alternative representations of the temporal evolution of the laser powder bed fusion (LPBF) process. In order to provide the reader with sufficient context for the following discussion, the modeling challenge and its underlying cause, as well as current attempts to provide a satisfactory solution, will be discussed. Next, the author will describe two numerical methods (multirate time integration and parallel-in-time) to better represent the temporal scales of the problem. Finally, the results of applying these methods to a two-dimensional version of the heat transfer portion of the LPBF problem will be presented, reporting on both convergence and performance behavior, which indicate the possibility of significant speedup of solution calculations (40–100 times), along with well-characterized effects on solution accuracy.

A New Class of High-Order Methods for Multirate Differential Equations

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified “fast” initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these multirate exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.

Multirate implicit Euler schemes for a class of differential–algebraic equations of index-1

Systems of differential equations which consist of subsystems with widely different dynamical behaviour can be integrated by multirate time integration schemes to increase the efficiency. These schemes allow the usage of inherent step sizes according to the dynamical properties of the subsystem. In this paper, we extend the multirate implicit Euler method to semi-explicit differential–algebraic equations of index-1 where the algebraic constraints only occur in the slow changing subsystem. We discuss different coupling approaches and show that consistency and convergence order 1 can be reached. Numerical experiments validate the analytical results.

Multi-rate time integration on overset meshes

Overset meshes are an effective tool for the computational fluid dynamic simulation of problems with complex geometries or multiscale spatio-temporal features. When the maximum allowable timestep on one or more meshes is significantly smaller than on the remaining meshes, standard explicit time integrators impose inefficiencies for time-accurate calculations by requiring that all meshes advance with the smallest timestep. With the targeted use of multi-rate time integrators, separate meshes can be time-marched at independent rates to avoid wasteful computation while maintaining accuracy and stability. This work applies time-explicit multi-rate integrators to the simulation of the compressible Navier-Stokes equations discretized on overset meshes using summation-by-parts (SBP) operators and simultaneous approximation term (SAT) boundary conditions. We introduce a class of multi-rate Adams-Bashforth (MRAB) schemes that offer significant stability improvements and computational efficiencies for SBP-SAT methods. We present numerical results that confirm the efficacy of MRAB integrators, outline a number of implementation challenges, and demonstrate a reduction in computational cost enabled by MRAB. We also investigate the use of our method in the setting of a large-scale distributed-memory parallel implementation where we discuss concerns involving load balancing and communication efficiency.

An enhanced transient solver with dynamic p‐adaptation and multirate time integration for electromagnetic and multiphysics simulations

AbstractThis paper presents an enhanced discontinuous Galerkin time‐domain (DGTD) solver for the simulation of both pure electromagnetic (EM) and multiphysics problems. By exploring the local variation of the physics, the polynomial expansion order of the physical quantity of interest is allowed to be adjusted locally and dynamically during the real time of a simulation, leading to a dynamic p‐adaptation technique that can provide a sufficient spatial resolution whenever and wherever needed while keeping the computational cost low. On top of the dynamic adaptation, the constraint of the time step size by an explicit time integration method is alleviated through the application of a multirate time integration (MTI) technique, which permits different time step sizes in elements with different sizes and polynomial orders. Together with the dynamic adaptation and MTI techniques, the enhanced DGTD method is both accurate and efficient in space and time and is employed in the numerical investigations of EM and multiphysics problems to demonstrate its performance and application.

A Class of Multirate Infinitesimal GARK Methods

Differential equations arising in many practical applications are characterized by multiple time scales. Multirate time integration seeks to solve them efficiently by discretizing each scale with a different, appropriate time step, while ensuring the overall accuracy and stability of the numerical solution. In a seminal paper Knoth and Wolke (APNUM, 1998) proposed a hybrid solution approach: discretize the slow component with an explicit Runge-Kutta method, and advance the fast component via a modified fast differential equation. The idea led to the development of multirate infinitesimal step (MIS) methods by Wensch et al. (BIT, 2009.)G\"{u}nther and Sandu (BIT, 2016) explained MIS schemes as a particular case of multirate General-structure Additive Runge-Kutta (MR-GARK) methods. The hybrid approach offers extreme flexibility in the choice of the numerical solution process for the fast component. This work constructs a family of multirate infinitesimal GARK schemes (MRI-GARK) that extends the hybrid dynamics approachin multiple ways. Order conditions theory and stability analyses are developed, and practical explicit and implicit methods of up to order four are constructed. Numerical results confirm the theoretical findings. We expect the new MRI-GARK family to be most useful for systems of equations with widely disparate time scales, where the fast process is dispersive, and where the influence of the fast component on the slow dynamics is weak.

An Advanced EM-Plasma Simulator Based on the DGTD Algorithm With Dynamic Adaptation and Multirate Time Integration Techniques

In a numerical simulation of the interactions between high-intensity electromagnetic (EM) waves and plasma fluids, a good numerical resolution is required for all the physical quantities of interest, especially the plasma density distribution around the edge of the plasma bulk. Since the plasma formation and evolution is a dynamic process, it is difficult to predetermine the appropriate resolution at a given location. In this paper, we describe a discontinuous Galerkin time-domain (DGTD) based dynamic adaptation algorithm, which is able to adjust either the local mesh size or the local polynomial order in real time of a simulation to provide a sufficient numerical resolution of the physics while keeping the total computational cost low. To alleviate the constraint of the time step size of an explicit time integrator, a multirate time integration method is employed to advance the physics in time, which permits the application of different time step sizes in elements with different sizes or polynomial orders. With these techniques, the DGTD simulation of EM–plasma interactions can achieve a good accuracy and high efficiency.

Design of High-Order Decoupled Multirate GARK Schemes

Multirate time integration methods apply different step sizes to resolve different components of a system based on local activity levels. This local selection of step sizes allows increased computational efficiency while achieving the desired solution accuracy. While the multirate idea is elegant and has been around for decades, multirate methods are not yet widely used in applications. This is due, in part, to the difficulties created by the construction of high-order multirate schemes. Seeking to overcome these challenges, this work focuses on the design of practical high-order multirate methods using the theoretical framework of multirate general-structure additive Runge--Kutta (MrGARK) methods [M. Günther and A. Sandu, Numer. Math., 133 (2016), pp. 497--524], which provides generic order conditions and linear and nonlinear stability analyses. A set of design criteria for practical multirate methods is defined herein: method coefficients should be generic in the step size ratio but should not depend strongly on this ratio, unnecessary coupling between the fast and the slow components should be avoided, and the step size controllers should adjust both the micro- and the macrosteps. Using these criteria, we develop MrGARK schemes of up to order four that are explicit-explicit (both the fast and slow component are treated explicitly), implicit-explicit (implicit in the fast component and explicit in the slow one), and explicit-implicit (explicit in the fast component and implicit in the slow one). Numerical experiments verify the accuracy of these new schemes.

An improved coupled thermo-mechanic bond-based peridynamic model for cracking behaviors in brittle solids subjected to thermal shocks

An improved coupled thermo-mechanical bond-based peridynamics with multirate time integration scheme is developed to study the thermal shock cracking behaviors in brittle solids. In the proposed numerical method, effects of the relative distance between material points on thermoelastic stiffness of a bond are considered in the deformed configuration. The periodical and hierarchical characteristics of thermal crack patterns of ceramics in quenching tests are accurately predicted in 2-D and 3-D cases. The present numerical results are in good agreement with the previous experimental observations and data. It is shown that the proposed numerical model is simple and efficient to simulate the periodical and hierarchical initiation and propagation of cracks in brittle solids under thermal shocks. The present numerical simulations provide direction observations on the whole processes of crack initiation and propagation, which is a quite difficult problem in the laboratory experiments.